Other than iodized salt, eggs and seafood, thyroid hormones(T3 & T4) contain iodine as part of their chemical structure. I'm no expert; just another seeker on the journey to improved health and function. Low iron or ferritin can negatively impact T4 to T3 conversion. Low cortisol the same thing. Low selenium, too. There are people who don't do well on T4, whether alone or in NDT. When I first was prescribed thyroid, it was Cytomel brand, which is T3.
As your free T3 and free T4 increase, your TSH will go down. Again, from what I've read, it sounds like either you're underdosed and your doctor is treating you by the labs or T4 is converting to RT3, blunting uptake of T3.
I just stopped T3/T4 drops because yesterday, I suddenly wasn't feeling well. Dry itchy skin, constipation, worsening food sensitivities, blunted mood. This morning started on T3 drops; 75 mcg, which is where I was at with the combo drops.